Finite Group Databasebeta

← #388499

Subgroups — #388499

Order 512.

1458 conjugacy classes of subgroups.

The subgroup lattice has 629 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23386C211yesmaximal, characteristic
2#21191C212yesmaximal, complemented
3C2×#1929C212yesmaximal, complemented
4C2×#1896C212yesmaximal, complemented
5#11097C211yesmaximal, characteristic, complemented
6#11068C212yesmaximal, complemented
7#11025C212yesmaximal, complemented
8#10887C212yesmaximal, complemented
9#10712C211yesmaximal, characteristic, complemented
10C22×#135C2211yescharacteristic
11C2×#525C2212yes—
12C23×C42C2211yescharacteristic
13C22×#108C2211yescharacteristic
14C2×#424C2212yes—
15C22×#104C2212yes—
16C22×#104C2212yes—
17C2×#416C2212yes—
18C2×#408C2212yes—
19C2×#408C2212yes—
20C2×#407C2212yes—
21C22×C22⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C2×#405C2212yes—
24#2001C411yescharacteristic, complemented
25#2001C411yescharacteristic, complemented
26#2000C412yescomplemented
27#2000C412yescomplemented
28#1999C411yescharacteristic, complemented
29#1999C411yescharacteristic, complemented
30#1955—22no—
31#1955—22no—
32#1954—22no—
33#1954—22no—
34#1929—28no—
35#1896—28no—
36C2×#341C2212yes—
37C2×#341C2211yescharacteristic
38#1735C2211yescharacteristic
39C2×#339C2212yes—
40C2×#339C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43C22×#41C2312yes—
44C24×C4—21no—
45C24×C4C2311yescharacteristic
46C2×#135—42no—
47C2×#135—42no—
48C2×#135—41no—
49C2×#135—41no—
50C22×Q16C2×C411yescharacteristic, complemented
51C22×SD16C2×C412yescomplemented
52C22×D8C2×C411yescharacteristic, complemented
53C22×M16C2312yes—
54C22×M16C2312yes—
55C23×C8C2311yescharacteristic
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58#525—24no—
59#525—24no—
60#525—42no—
61#525—42no—
62C2×#123—22no—
63C2×C22⋊Q8C2×C412yes—
64C2×C22⋊Q8C2×C412yes—
65C2×#121C2×C412yes—
66C2×#121C2×C412yes—
67C2×#120—22no—
68C2×C4×D4—22no—
69C2×C4×D4—22no—
70C2×#117C2312yes—
71C22×C42—24no—
72C22×C42—21no—
73C22×C42—41no—
74C22×C42—41no—
75C22×C42—42no—
76C22×C42C2312yes—
77C2×#108—24no—
78C2×#108—41no—
79#424—24no—
80#424—42no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×#104—24no—
84C2×#104—24no—
85C2×#104—24no—
86C2×#104—42no—
87C2×#104—42no—
88C2×#104—42no—
89#416—24no—
90#416—24no—
91C2×#103C2×C412yescomplemented
92C2×#103C2×C412yescomplemented
93C2×#102C2×C412yescomplemented
94C2×#102C2×C412yescomplemented
95#408—24no—
96#408—42no—
97#408—42no—
98#408—42no—
99#407—24no—
100#407—42no—
101C2×C22⋊C8—22no—
102C2×C22⋊C8—22no—
103C2×C22⋊C8—22no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×C22⋊C8—22no—
107C2×C22⋊C8—41no—
108C2×C22⋊C8—41no—
109C2×C22⋊C8—41no—
110C2×C22⋊C8—41no—
111C2×C22⋊C8C2×C411yescharacteristic
112C2×C22⋊C8C2×C411yescharacteristic
113C2×C22⋊C8C2×C411yescharacteristic
114C2×C22⋊C8C2×C411yescharacteristic
115C2×C22⋊C8D411yescharacteristic
116C2×C22⋊C8D412yes—
117C2×C22⋊C8D411yescharacteristic
118#405—24no—
119#405—42no—
120C2×#97—22no—
121C2×C4×C8—22no—
122#341—42no—
123#341—41no—
124#340C2×C411yescharacteristic, complemented
125#340C2×C411yescharacteristic, complemented
126#340C2×C412yescomplemented
127#339—24no—
128#339D418yescomplemented
129#336—24no—
130#336—42no—
131C25#4111yescharacteristic
132C2×#41—22no—
133C2×#41—22no—
134C2×#41—22no—
135C2×#41—22no—
136C2×#41—22no—
137C2×#41—22no—
138C2×#41—22no—
139C2×#41—22no—
140C2×#41—22no—
141C2×#41—22no—
142C2×#41—22no—
143C2×#41—22no—
144C22×Q8C22×C412yes—
145C22×D4—22no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C22×C412yes—
149C23×C4—22no—
150C23×C4—22no—
151C23×C4—22no—
152C23×C4—22no—
153C23×C4—22no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—24no—
159C23×C4—22no—
160C23×C4—44no—
161C23×C4—44no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4C22×C411yescharacteristic
165C23×C4C22×C411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4#4111yescharacteristic
169C23×C4C2411yescharacteristic
170#135—48no—
171#135—44no—
172#135—44no—
173C2×Q16—41no—
174C2×Q16—42no—
175C2×SD16—42no—
176C2×SD16—42no—
177C2×SD16—42no—
178C2×D8—41no—
179C2×D8—42no—
180C2×M16—22no—
181C2×M16—22no—
182C2×M16—22no—
183C2×M16—22no—
184C2×M16—22no—
185C2×M16—22no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—22no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16C2×D414yes—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—22no—
202C22×C8—21no—
203C22×C8—22no—
204C22×C8—22no—
205C22×C8—22no—
206C22×C8—21no—
207C22×C8—22no—
208C22×C8C22×C411yescharacteristic
209C22×C8C22×C411yescharacteristic
210C22×C8C22×C411yescharacteristic
211C22×C8C22×C411yescharacteristic
212C22×C8C2×D412yes—
213C22×C8C2×D412yes—
214#123—44no—
215#123—44no—
216C22⋊Q8—24no—
217C22⋊Q8—24no—
218C22⋊Q8—42no—
219C22⋊Q8—42no—
220#121—24no—
221#121—24no—
222#121—42no—
223#121—42no—
224#120—44no—
225#120—44no—
226C4×D4—44no—
227C4×D4—44no—
228C4×D4—44no—
229C4×D4—44no—
230#117—22no—
231#117—22no—
232#117C2×D414yes—
233#117#4114yes—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—22no—
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C22×C412yes—
238C2×C22⋊C4—22no—
239C2×C22⋊C4—22no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C42—22no—
244C2×C42—22no—
245C2×C42—22no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—44no—
250C2×C42—48no—
251C2×C42—44no—
252C2×C42—42no—
253C2×C42—44no—
254C2×C42C2×D414yes—
255#108—44no—
256C4⋊C8—44no—
257C4⋊C8—44no—
258#104—48no—
259#104—48no—
260#103—24no—
261#103—24no—
262#102—24no—
263#102—24no—
264C22⋊C8—41no—
265C22⋊C8—41no—
266C22⋊C8—41no—
267C22⋊C8—41no—
268C22⋊C8—41no—
269C22⋊C8—42no—
270C22⋊C8—41no—
271C22⋊C8—44no—
272C22⋊C8—44no—
273C22⋊C8—44no—
274C22⋊C8—44no—
275C22⋊C8—44no—
276C22⋊C8—44no—
277#97—44no—
278C4×C8—44no—
279C24—21no—
280C24—21no—
281C24—22no—
282C24—22no—
283C24—22no—
284C24—22no—
285C24—22no—
286C24—42no—
287C24C4×D411yescharacteristic
288C24C4×D411yescharacteristic
289C24C2×#4111yescharacteristic
290#41—42no—
291#41—42no—
292#41—42no—
293#41—42no—
294#41—42no—
295#41—42no—
296#41—42no—
297#41—42no—
298#41—42no—
299#41—42no—
300#41—42no—
301#41—42no—
302#41—42no—
303#41—42no—
304#41—42no—
305#41—42no—
306C2×Q8—22no—
307C2×Q8—22no—
308C2×Q8—22no—
309C2×Q8—22no—
310C2×Q8—22no—
311C2×Q8—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—44no—
319C2×D4—44no—
320C2×D4—42no—
321C2×D4—42no—
322C2×D4—42no—
323C2×D4—42no—
324C2×D4—42no—
325C2×D4—42no—
326C2×D4—44no—
327C2×D4—44no—
328C2×D4—82no—
329C2×D4—82no—
330C22×C4—24no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—21no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—21no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—22no—
354C22×C4—22no—
355C22×C4—22no—
356C22×C4—416no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—48no—
360C22×C4—48no—
361C22×C4—416no—
362C22×C4—48no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—41no—
370C22×C4—42no—
371C22×C4—41no—
372C22×C4—42no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—42no—
380C22×C4—42no—
381C22×C4C4×D411yescharacteristic
382C22×C4C4×D411yescharacteristic
383C22×C4C23×C411yescharacteristic
384C22×C4C22×D412yes—
385C22×C4C22×D411yescharacteristic
386C22×C4C2×#4112yes—
387C22×C4H(5,𝔽2)11yescharacteristic
388C22×C4H(5,𝔽2)12yes—
389C22×C4H(5,𝔽2)12yes—
390C22×C4H(5,𝔽2)11yescharacteristic
391C22×C4H(5,𝔽2)11yescharacteristic
392C22×C42−1+411yescharacteristic
393Q16—44no—
394SD16—48no—
395D8—44no—
396M16—24no—
397M16—24no—
398M16—42no—
399M16—42no—
400M16—42no—
401M16—42no—
402M16—42no—
403M16—42no—
404C2×C8—22no—
405C2×C8—22no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—24no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—42no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—41no—
424C2×C8—42no—
425C2×C8—42no—
426C2×C8—41no—
427C2×C8—42no—
428C2×C8—42no—
429C2×C8—41no—
430C2×C8—41no—
431C2×C8—42no—
432C2×C8—42no—
433C2×C8—42no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—44no—
437C4⋊C4—44no—
438C4⋊C4C4×D414yes—
439C4⋊C4C4×D414yes—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—44no—
443C22⋊C4—44no—
444C22⋊C4—44no—
445C22⋊C4—44no—
446C42—24no—
447C42—24no—
448C42—24no—
449C42—24no—
450C42—48no—
451C42—42no—
452C42—42no—
453C23—21no—
454C23—21no—
455C23—22no—
456C23—24no—
457C23—24no—
458C23—21no—
459C23—21no—
460C23—22no—
461C23—22no—
462C23—21no—
463C23—21no—
464C23—22no—
465C23—22no—
466C23—44no—
467C23—44no—
468C23—44no—
469C23—44no—
470C23—42no—
471C23—42no—
472C23—41no—
473C23—42no—
474C23—41no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23#44312yes—
483C23#44412yes—
484C23C2×C4×D411yescharacteristic
485C23#53811yescharacteristic
486C23#53911yescharacteristic
487C23#54612yes—
488C23#55112yes—
489Q8—42no—
490Q8—42no—
491Q8—42no—
492Q8—42no—
493D4—42no—
494D4—42no—
495D4—42no—
496D4—42no—
497D4—88no—
498D4—88no—
499D4—82no—
500D4—82no—
501D4—82no—
502D4—82no—
503C2×C4—28no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—432no—
516C2×C4—44no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—41no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—82no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4C2×C4×D412yes—
544C2×C4#51812yes—
545C2×C4#51811yescharacteristic
546C2×C4#52011yescharacteristic
547C2×C4#53411yescharacteristic
548C2×C4#53512yes—
549C2×C4#53712yes—
550C2×C4#53711yescharacteristic
551C2×C4#54012yes—
552C2×C4#54212yes—
553C2×C4D4211yescharacteristic
554C2×C4D4212yes—
555C2×C4#54612yes—
556C2×C4#55012yes—
557C2×C4#55911yescharacteristic
558C2×C4#56012yes—
559C2×C4#56011yescharacteristic
560C2×C4H(3,𝔽4)11yescharacteristic
561C8—41no—
562C8—41no—
563C8—41no—
564C8—41no—
565C8—42no—
566C8—42no—
567C8—44no—
568C8—82no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—21no—
574C22—21no—
575C22—21no—
576C22—21no—
577C22—24no—
578C22—24no—
579C22—22no—
580C22—22no—
581C22—44no—
582C22—48no—
583C22—44no—
584C22—42no—
585C22—42no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—41no—
590C22—82no—
591C22—82no—
592C22—82no—
593C22—82no—
594C22—82no—
595C22—82no—
596C22—82no—
597C22#232511yescharacteristic
598C22#235812yes—
599C22#267011yescharacteristic
600C22#270512yes—
601C22#275211yescharacteristic
602C22C2×#44311yescharacteristic
603C22C2×#44411yescharacteristic
604C22#295612yes—
605C4—48no—
606C4—44no—
607C4—44no—
608C4—41no—
609C4—41no—
610C4—82no—
611C4#232412yes—
612C4#236611yescharacteristic
613C4#237511yescharacteristic
614C4#267312yes—
615C4#267511yescharacteristic
616C4#271211yescharacteristic
617C2—21no—
618C2—22no—
619C2—21no—
620C2—42no—
621C2—42no—
622C2—162no—
623C2#1452111yescharacteristic
624C2#2220411yescharacteristic
625C2#2220711yescharacteristic
626C2#2288612yes—
627C2#2288812yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radical#23386C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #21191C2semidirect
3. C2×#1929C2semidirect
4. C2×#1896C2semidirect
5. #11097C2semidirect
6. #11068C2semidirect
7. #11025C2semidirect
8. #10887C2semidirect
9. #10712C2semidirect
24. #2001C4semidirect
25. #2001C4semidirect
26. #2000C4semidirect
27. #2000C4semidirect
28. #1999C4semidirect
29. #1999C4semidirect
50. C22×Q16C2×C4semidirect
51. C22×SD16C2×C4semidirect
52. C22×D8C2×C4semidirect
91. C2×#103C2×C4semidirect
92. C2×#103C2×C4semidirect
93. C2×#102C2×C4semidirect
94. C2×#102C2×C4semidirect
124. #340C2×C4semidirect
125. #340C2×C4semidirect
126. #340C2×C4semidirect
128. #339D4semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#233862
#211912
C2×#19292
C2×#18962
#110972
#110682
#110252
#108872
#107122